Quadrature (volume) of the pyramid
One third of the height times the base, shown with a prism and pyramid figures
Folio 34v states the rule for the quadrature (volume) of a pyramid or cone: it equals one third of its height multiplied by the whole area of its base. Leonardo works an example, an axis of 3 braccia and a base of 4 square braccia giving a volume of 4, and draws a wireframe prism with an inscribed pyramid, with a smaller pyramid at the right, to show the one-third relation to the enclosing cylinder. The remainder of the sheet is filled with the supporting multiplications and divisions.
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The volume of a pyramid equals one third of its height times the base area
The general rule is set down twice: multiplying a third of the height of the pyramid by the whole base gives the quadrature of the whole pyramid, and the quadrature of every pyramid lies in a third of its height multiplied by the whole surface of the base.
Worked example: axis 3, base 4 square braccia; pyramid inscribed in its cylinder
To square the pyramid one takes a third of the height of its cylinder, which is equal in quantity to the pyramid; with an axis of 3 braccia and a base of 4 square braccia, one third of the axis (1) times 4 gives 4. A wireframe prism with an inscribed pyramid drawn on the sheet illustrates the relation.
Supporting multiplications and divisions
The lower half of the sheet carries the calculations behind the estimates, with long products and divisions (58880, 60000, 6444 recurring as divisors) and mixed fractions reduced to canne. They continue the fortification arithmetic of the preceding leaves.